Actually, I am not sure the "speaker" of this poem is the women in the poem in a romantic relationship. Rather, I think the speaker in the poem is the poet herself, describing a situation in which a woman is involved in a romantic relationship.
The poet's use of the words associated with housekeeping is precisely what makes the poem so interesting, so effective in getting its message across. The housekeeping words, such as: "the taps less vocal/the panes relieved of grime...a piano with a Persian shawl, a cat..." all combine to add a distinctively unromantic relationship, or perhaps a relationship cluttered with household goods -- no romance!
The unromanticism is complete when Adrinenne Rich introduces the object of the woman's love:
Meanwhile, he, with a yawn,
sounded a dozen notes upon the keyboard,
declared it out of tune, shrugged at the mirror,
rubbed at his beard, went out for cigarettes...
The reader could have expected the man to wake up, get out of the bed, come up and hold the woman affectionately from behind and, with a kiss on the nape of her neck, wish his beloved a good morning; but, no, he simply declares the piano out of tune, rubs at his beard and goes out for cigarettes! While the woman busies herself with the house chores, alone in the studio apartment which was supposed to be a temple of love!
The title of the poem, thus, --"living in sin,"-- is replete with irony. The phase is supposed to mean a couple of the opposite sex are living together like husband and wife (or lovers) without getting married (hence, "sin!") People usually do this when they are in love. But the woman keeps moving back and forth in her mind, in and out of romance. By day light, it seems, love dies; but in the evening -- "she'is back in love!
The household goods described in the poem are supposed to represent the dark side of a relationship gone stale. "Not only is there no love," as a the poet Alan Ginsberg wrote in a poem in "Howl," of love gone stale, "there is responsibility!"
I think that is the irony of this poem.
Do remember to check out enotes on this poem.